I am confused between total derivatives and total differential. What is the difference between total derivatives and total differential?
-
$\begingroup$ For a question like this, you should provide a definition of each of these terms to get a good answer. $\endgroup$ – Bernard W Jul 6 '17 at 1:04
the total differential is
$dz = \frac {\partial z}{\partial x} dx + \frac {\partial z}{\partial y} dy$
How much do we expect $z$ to change for some changes in $x$ and $y?$
The total derivative is
$\frac {dz}{dt} = \frac {\partial z}{\partial x} \frac {dx}{dt} + \frac {\partial z}{\partial y} \frac {dy}{dt}$
$x,y$ are both functions with respect to parameter $t$ what is the derivative of $z$ with respect to this parameter?
-
1$\begingroup$ Notice that in the equation for the total derivative, two different functions are both being called $z$. This is a common abuse of notation, but I think it sometimes causes confusion. $\endgroup$ – littleO Jul 6 '17 at 2:49
Let $f: U \subset \Bbb R^n \to \Bbb R^m$ be differentiable.
The total derivative of $f$ at $a$ is the linear map $df_a$ such that $f(a+t) - f(a) = df_a(t) + o(t)$.
For $m=1$, the total differential of $f$ is
$$df = \sum_{i=1}^m \frac{\partial f}{\partial x_i} dx_i$$
Hope this helps.